Optimal. Leaf size=119 \[ -\frac {a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac {\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac {a \sin ^3(c+d x)}{3 b^2 d}-\frac {\sin ^4(c+d x)}{4 b d} \]
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Rubi [A] time = 0.17, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ -\frac {\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac {a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac {a \sin ^3(c+d x)}{3 b^2 d}-\frac {\sin ^4(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )}{b^2 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (b^2-x^2\right )}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 \left (1-\frac {b^2}{a^2}\right )-\left (a^2-b^2\right ) x+a x^2-x^3+\frac {a^2 \left (-a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {a^2 \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac {a \left (a^2-b^2\right ) \sin (c+d x)}{b^4 d}-\frac {\left (a^2-b^2\right ) \sin ^2(c+d x)}{2 b^3 d}+\frac {a \sin ^3(c+d x)}{3 b^2 d}-\frac {\sin ^4(c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 104, normalized size = 0.87 \[ \frac {6 b^2 \left (b^2-a^2\right ) \sin ^2(c+d x)+12 a b \left (a^2-b^2\right ) \sin (c+d x)+12 a^2 \left (b^2-a^2\right ) \log (a+b \sin (c+d x))+4 a b^3 \sin ^3(c+d x)-3 b^4 \sin ^4(c+d x)}{12 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 97, normalized size = 0.82 \[ -\frac {3 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 12 \, {\left (a^{4} - a^{2} b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 2 \, a b^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 117, normalized size = 0.98 \[ -\frac {\frac {3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b \sin \left (d x + c\right )^{2} - 6 \, b^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) + 12 \, a b^{2} \sin \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{4} - a^{2} b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 144, normalized size = 1.21 \[ -\frac {\sin ^{4}\left (d x +c \right )}{4 b d}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3 b^{2} d}-\frac {a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d \,b^{3}}+\frac {\sin ^{2}\left (d x +c \right )}{2 b d}+\frac {\sin \left (d x +c \right ) a^{3}}{d \,b^{4}}-\frac {a \sin \left (d x +c \right )}{b^{2} d}-\frac {a^{4} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{5}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 105, normalized size = 0.88 \[ -\frac {\frac {3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \, {\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{4} - a^{2} b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.62, size = 107, normalized size = 0.90 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^4}{4\,b}-{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{2\,b}-\frac {a^2}{2\,b^3}\right )-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3\,b^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^4-a^2\,b^2\right )}{b^5}+\frac {a\,\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2}{b^3}\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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